3-D hydrodynamic model using the spectral method in the vertical direction for bend flow simulations
YANG Fei, FU Xudong
Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China
Abstract: The traditional depth-averaged 2-D hydrodynamic models for bend flow simulations assume velocity profiles for the secondary flow terms in the momentum equations which are unable to adjust to dynamic conditions. More flexible three-dimensional models are not very efficient. A simplified 3-D model using a spectral method in the vertical direction was developed with the flow velocity components modeled by orthogonal polynomials in the vertical direction using polynomial coefficient equations obtained using the weighted residuals method with the advection terms defined at the vertical Gauss points by the semi-Lagrangian scheme. Simulated flow structures in a sharp bend open channel match well with measured data for polynomials having degrees larger than 1 with reasonable flow structures. The mean error of the predicted main flow location is less than 7%, equivalent to other 3-D hydrodynamic models. The eddy viscosity is solved in a simple way with consideration of the turbulence anisotropy between the vertical and horizontal directions. Since this method does not have a vertical grid, the calculational efficiency is proved to be to 2-D models.
Key words: spectral method     3-D hydrodynamic model     secondary flow     bend flow     anisotropy

1 模型建立 1.1 控制方程

 $\frac{\partial }{{\partial \xi }}\frac{u}{J} + \frac{\partial }{{\partial \eta }}\frac{v}{J} + \frac{\partial }{{\partial z}}\frac{w}{J} = 0.$ (1)

 $\begin{array}{l} \frac{{\partial u}}{{\partial t}} + J\left( {\frac{\partial }{{\partial \xi }}\frac{{uu}}{J} + \frac{\partial }{{\partial \eta }}\frac{{uv}}{J} + \frac{\partial }{{\partial z}}\frac{{uw}}{J}} \right) = \\ - {\alpha _1}uu - {\alpha _2}uv - {\alpha _3}vv - \\ g\left( {{\beta _1}\frac{{\partial H}}{{\partial \xi }} + {\beta _2}\frac{{\partial H}}{{\partial \eta }}} \right) + \frac{\partial }{{\partial z}}\left( {{v_{\rm{z}}}\frac{{\partial u}}{{\partial z}}} \right) + \\ \frac{\partial }{{\partial \xi }}\left( {{v_{\rm{h}}}\xi _r^2\frac{{\partial u}}{{\partial \xi }}} \right) + \frac{\partial }{{\partial \eta }}\left( {{v_{\rm{h}}}\eta _r^2\frac{{\partial u}}{{\partial \eta }}} \right), \end{array}$ (2a)
 $\begin{array}{l} \frac{{\partial v}}{{\partial t}} + J\left( {\frac{\partial }{{\partial \xi }}\frac{{uv}}{J} + \frac{\partial }{{\partial \eta }}\frac{{vv}}{J} + \frac{\partial }{{\partial z}}\frac{{vw}}{J}} \right) = \\ - {\alpha _4}uu - {\alpha _5}uv - {\alpha _6}vv - \\ g\left( {{\beta _3}\frac{{\partial H}}{{\partial \xi }} + {\beta _4}\frac{{\partial H}}{{\partial \eta }}} \right) + \frac{\partial }{{\partial z}}\left( {{v_{\rm{z}}}\frac{{\partial v}}{{\partial z}}} \right) + \\ \frac{\partial }{{\partial \xi }}\left( {{v_{\rm{h}}}\xi _r^2\frac{{\partial v}}{{\partial \xi }}} \right) + \frac{\partial }{{\partial \eta }}\left( {{v_{\rm{h}}}\eta _r^2\frac{{\partial v}}{{\partial \eta }}} \right). \end{array}$ (2b)

 $\begin{array}{l} {\beta _1} = \xi _x^2 + \xi _y^2,{\beta _2} = {\xi _x}{\eta _x} + {\xi _y}{\eta _y},\\ {\beta _3} = {\xi _x}{\eta _x} + {\xi _y}{\eta _y},{\beta _4} = \eta _x^2 + \eta _y^2. \end{array}$ (3a)

 $\begin{array}{l} {\alpha _1} = {\xi _x}\frac{{{\partial ^2}x}}{{\partial {\xi ^2}}} + {\xi _y}\frac{{{\partial ^2}y}}{{\partial {\xi ^2}}},{\alpha _2} = 2\left( {{\xi _x}\frac{{{\partial ^2}x}}{{\partial \xi \partial \eta }} + {\xi _y}\frac{{{\partial ^2}y}}{{\partial \xi \partial \eta }}} \right),\\ {\alpha _3} = {\xi _x}\frac{{{\partial ^2}x}}{{\partial {\eta ^2}}} + {\xi _y}\frac{{{\partial ^2}y}}{{\partial {\eta ^2}}},{\alpha _4} = {\eta _x}\frac{{{\partial ^2}x}}{{\partial {\xi ^2}}} + {\eta _y}\frac{{{\partial ^2}y}}{{\partial {\xi ^2}}},\\ {\alpha _5} = 2\left( {{\eta _x}\frac{{{\partial ^2}x}}{{\partial \xi \partial \eta }} + {\eta _y}\frac{{{\partial ^2}y}}{{\partial \xi \partial \eta }}} \right),{\alpha _6} = {\eta _x}\frac{{{\partial ^2}x}}{{\partial {\eta ^2}}} + {\eta _y}\frac{{{\partial ^2}y}}{{\partial {\eta ^2}}}. \end{array}$ (3b)

 $\begin{array}{l} u \approx \sum\limits_{i = 0}^N {{c_i}{p_i}} ,v \approx \sum\limits_{i = 0}^N {{d_i}{p_i}} ,\\ {v_{\rm{z}}} \approx \sum\limits_{i = 0}^N {{e_{{\rm{z}}\;i}}{p_i}} ,{v_{\rm{h}}} \approx \sum\limits_{i = 0}^N {{e_{{\rm{h}}i}}{p_i}} . \end{array}$ (4)

1.2 加权余量法

 $\begin{array}{l} \int_{{z_{\rm{b}}}}^H {{u^{n + 1}}{p_k}{\rm{d}}z} = \int_{{z_{\rm{b}}}}^H {{u^{ * ;n}}{p_k}{\rm{d}}z} + \Delta t\int_{{z_{\rm{b}}}}^H {\left( {{f^n} + D_{\rm{h}}^n} \right){p_k}{\rm{d}}z} + \\ {p_k}\Delta t{v_{\rm{z}}}\frac{{\partial {u^n}}}{{\partial z}}\left| {_{{z_{\rm{b}}}}^H} \right. - \Delta t\int_{{z_{\rm{b}}}}^H {\frac{{\partial {p_k}}}{{\partial z}}{v_{\rm{z}}}\frac{{\partial {u^n}}}{{\partial z}}{\rm{d}}z} ,k = 0, \cdots ,N. \end{array}$ (5)

 $\begin{array}{*{20}{c}} {\frac{h}{2}\sum\limits_{j = 1}^N {{c_j}^{n + 1}{B_{jk}}} = \frac{h}{2}\sum\limits_{j = 1}^N {{c_{j * }}^n{B_{jk}}} + \Delta t\frac{h}{2}{f^n}{B_{0k}} + }\\ {\frac{\partial }{{\partial \xi }}\left( {\sum\limits_{m = 0}^N {{e_{{\rm{h}}m}}\xi _r^2\frac{\partial }{{\partial \xi }}} \sum\limits_{j = 0}^N {{c_j}^n} } \right){C_{jmk}} + }\\ {\frac{\partial }{{\partial \eta }}\left( {\sum\limits_{m = 0}^N {{e_{{\rm{h}}m}}\eta _r^2\frac{\partial }{{\partial \eta }}} \sum\limits_{j = 0}^N {{c_j}^n} } \right){C_{jmk}} + \varphi \Delta t{\nu _{\rm{z}}}\frac{{\partial {u^n}}}{{\partial z}}\left| {_{{z_{\rm{b}}}}^H} \right. - }\\ {\Delta t\frac{2}{h}\sum\limits_{m = 0}^N {{e_{{\rm{z}}m}}} \sum\limits_{j = 0}^N {{c_j}^n{D_{jk,m}}} ,\;\;\;\;k = 0,1, \cdots ,N.} \end{array}$ (6)

 $\begin{array}{l} {B_{jk}} = \int_{ - 1}^1 {{p_j}{p_k}{\rm{d}}\zeta } ,\\ {C_{jmk}} = \int_{ - 1}^1 {{p_k}{p_m}{p_j}{\rm{d}}\zeta } ,\\ {D_{jk,m}} = \int_{ - 1}^1 {\frac{{\partial {p_k}}}{{\partial \zeta }}{p_m}\frac{{\partial {p_j}}}{{\partial \zeta }}{\rm{d}}\zeta } . \end{array}$ (7)

 ${v_{\rm{z}}}\frac{{\partial u}}{{\partial z}}\left| {_{{z_{\rm{b}}}}} \right. = {U_ * }{u_ * } = {C_{\rm{d}}}{U_{\rm{b}}}{u_{\rm{b}}}.$ (8)

1.3 对流项

 $\begin{array}{l} \int_{ - 1}^1 {{p_i}J\left( {\frac{\partial }{{\partial \xi }}\frac{{uu}}{J} + \frac{\partial }{{\partial \eta }}\frac{{uv}}{J} + \frac{\partial }{{\partial z}}\frac{{uw}}{J}} \right){\rm{d}}\zeta } \approx \\ \sum\limits_{j = 1}^N {{{\left[ {{\omega _j}{p_i}J\left( {\frac{\partial }{{\partial \xi }}\frac{{uu}}{J} + \frac{\partial }{{\partial \eta }}\frac{{uv}}{J} + \frac{\partial }{{\partial z}}\frac{{uw}}{J}} \right)} \right]}_{\zeta = {\zeta _j}}}} . \end{array}$ (9)

 $- J\left( {\frac{\partial }{{\partial \xi }}\frac{{uu}}{J} + \frac{\partial }{{\partial \eta }}\frac{{uv}}{J} + \frac{\partial }{{\partial z}}\frac{{uw}}{J}} \right) \approx \frac{{{u^{ * ;n}} - {u^n}}}{{\Delta t}}.$ (10)

1.4 水位求解

 $\frac{\partial }{{\partial t}}\left( {\frac{h}{J}} \right) + \frac{\partial }{{\partial \xi }}\left( {\frac{{h\bar u}}{J}} \right) + \frac{\partial }{{\partial \eta }}\left( {\frac{{h\bar v}}{J}} \right) = 0.$ (11)

1.5 数值算法

2 急弯水槽试验验证

 图 1 文[3]的试验水槽形态

 ${\nu _{\rm{h}}} = 2{\nu _{\rm{z}}} = \frac{3}{2}{{\bar \nu }_{\rm{h}}}\left( {1.01 - {\zeta ^2}} \right),$ (12)
 ${{\bar \nu }_{\rm{h}}} = \frac{\kappa }{6}h{u_ * }.$ (13)

 阶数/其他模型 CPU用时/s 横向网格 垂向网格或Gauss点数 主流线位置均差/% S90中线流速均差/(m·s-1) S180中线流速均差/(m·s-1) 纵向 横向 纵向 横向 0 100 20 1 29.2 1 189 20 2 17.3 2 294 20 3 4.8 0.030 0.015 0.029 0.035 3 459 20 4 4.5 0.031 0.012 0.036 0.039 4 644 20 5 5.4 0.028 0.013 0.035 0.041 5 876 20 6 6.3 0.027 0.012 0.035 0.038 6 1 148 20 7 5.2 0.025 0.012 0.035 0.038 Z-R 101 35 14.7 vB-R 192 24 8.6 vB-L 192 24 6.6

 图 2 (网络版彩图)弯道垂向平均流速实测与模拟结果

 图 3 (网络版彩图)水位平面分布图

 图 4 主流线横向相对位置的沿程变化

 图 5 (网络版彩图)断面纵向流速分布实测与N=6模拟对比

 图 6 S90断面中心线纵向(右侧)与横向(左侧)流速垂向分布的模拟与实测值

 图 7 S180断面中心线纵向(右侧)与横向(左侧)流速垂向分布的模拟与实测值

3 讨论

4 结论

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